Modeling Water Flow Through Carbon Nanotube Membranes withEntrance/Exit EffectsMyung Eun Suka and Narayana R. Alurub

aNew Transportation Research Center, Korea Railroad Research Institute, 176, Cheldobangmulgwan-ro, Uiwang-si,Gyeonggi-do, South Korea; bDepartment of Mechanical Science and Engineering, Beckman Institute for AdvancedScience and Technology, University of Illinois at Urbana–Champaign, Urbana, Illinois, USA

ABSTRACTCarbon nanotube–based membranes have gained significant attention dueto their transport efficiency and wide range of applications, including mole-cular sieving and sensing. Recently, in order to attain high transport rates,many studies have focused on reducing membrane thickness. A reduction inmembrane thickness results in the dominance of entrance/exit effects oversurface effects, particularly for carbon nanotubes (CNTs), due to their hydro-phobicity. However, experimentally obtained nanoscale flow rate data span awide range, and entrance/exit effects are often neglected when analyzingthese data. In this study, we modeled the water flow rate through variouslengths and radii of CNTs using molecular dynamics simulations while alsotaking entrance/exit effects into consideration. Based on viscosity and sliplength calculations, a water flow model is proposed that covers variouslengths and radii of CNTs. Moreover, the enhancement factor of CNT mem-branes is reassessed using entrance/exit effects. The results of this study canbe used for the optimal design of ultraefficient CNT membranes for potentialapplications such as water filtration.

ARTICLE HISTORYReceived 11 February 2017Accepted 12 July 2017

KEYWORDSCarbon nanotubemembrane; membranepermeability; water flow;entrance and exit effect

Introduction

Studies on water flow through carbon nanotubes (CNTs) have been frequently conducted because ofthe high rate of water flow through them [1–3], their ion selectivity [4–6], and other factors [7–9]. In2006, Holt et al. successfully grew a CNT forest with a diameter of less than 2 nm to produce a CNTmembrane [2]. The CNT membrane showed about 100 to 1,000 times higher water permeabilitythan existing polymeric membranes [2]. Thereafter, many studies were conducted using moleculardynamic simulations, including observations of water flow through [10] CNTs and between [11]CNTs, discussions of the main factors and mechanisms governing fast water flow [12], and studieson ion rejection according to CNT diameter and the resulting possibility of their application fordesalination [4].

CNT membranes with various diameters and matrix fillers have been fabricated experimentally[7, 8, 13–15]. Baek et al. fabricated a CNT membrane with a diameter of ~4.8 nm using epoxy filler[8]. Recently, CNT–parylene membranes with a diameter of ~3.3 nm were studied by Bui et al. [7].CNT–wall membranes consisting only of CNTs were fabricated by Lee et al. [15]. Throughmechanical densification, the CNT–wall membranes can be produced with 6-nm-wide inner poresand 7-nm-wide outer pores [15]. The porosity is greatly increased in this case, compared to previous/existing CNTs filled with a polymer [7] or an Si3N4 matrix [2], and the permeability is also increased

CONTACT Myung Eun Suk msuk2@krri.re.kr New Transportation Research Center, Korea Railroad Research Institute, 176,Cheldobangmulgwan-ro, Uiwang-si, Gyeonggi-do 16105, South Korea.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umte.© 2017 Taylor & Francis

NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING2017, VOL. 21, NO. 4, 247–262https://doi.org/10.1080/15567265.2017.1355949

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by over 10 times. In addition, it has been shown that the permeability is further increased by makingthe entrances of the CNTs hydrophilic via an oxygen-reactive ion etching process [15].

Due to the fast water flow rate through CNTs [1–3], the gradually decreasing manufacturingcosts, and a greater feasibility of commercialization [16], the chances of CNT membranes replacingexisting filtration membranes have risen [17]. Depending on the size of the membrane, not onlywater but also protons [18, 19], ions [20], biomolecules [21], and other molecules can be transportedhighly efficiently. Therefore, the applications of CNT membranes are being expanded to include avariety of fields, including fuel cells [18], drug delivery [21], ion separation [22] and supercapacitors[23, 24]. For expanded applications of CNT membranes and the interpretation of various experi-mental results, an accurate comprehension is needed of the relationship between the mass transferrate and the dimensions of a CNT, particularly its radius and length. In experimental setups, the finecontrol of CNT size and the accurate measurement of the physical properties of the fluid inside theCNT and its mass transfer rate are difficult to achieve [25]. Therefore, molecular dynamics simula-tion is a useful tool for a comprehensive understanding of the matter.

There have been previous studies investigating the effects of CNT dimensions on the water flow rateusing molecular dynamics (MD) simulations [26–30]. Su and Yang [26] and Su and Guo [30] tested(6,6)–(15,15) CNTs, corresponding to radii of 0.41~1.01 nm, for a given CNT length of 2.56 nm. For agiven CNT radius of 0.54 nm, CNT length varied from 1.34 to 9.89 nm [26, 30]. The unidirectional waterflow rate was calculated by applying the pressure gradient, and changes in the flow rate with CNT radiusfor a given CNT length were described by an exponential function. As noted in Su and Guo [30], theexponential dependency of water flow rate on CNT radius is not expected from the continuum relation-ship (Hagen-Poiseuille relationship), which is scaled by Q~R4 and Q~L−1. Their studied CNT radiusranged from 0.41 to 1.01 nm, mostly corresponding to the subcontinuum regime [29]. For a (7,7) CNTwith a radius of 0.48 nm, Nicholls et al. [27] tested the effects of varying lengths on water flow rate; theyobserved an invariant water flux with changing lengths of the CNT. In this subcontinuum regime, watermolecules are transported by collective motion rather than by frictional flow [31]. The energy barrier atthe entrance determines the water transport rate, and hydrophilic pores provide a higher transport ratethan hydrophobic pores by lowering the energy barrier at the entrance [10].

The transition from the continuum to the subcontinuum regime of water flow through CNTs wasstudied in detail by Thomas and McGaughey [28]. By comparing the MD simulation results to amodified Hagen-Poiseuille (HP) relationship with a slip boundary condition and reduced waterviscosity inside the CNT, the transition was reported to occur at R ~ 0.7 nm [29]. For a CNT radiussmaller than ~0.7 nm, a deviation in the MD resulting from the modified HP relationship was found.Thomas and McGaughey [29] tested CNTs with two different lengths (75 and 150 nm) to eliminatethe pressure loss at the entrance and exit regions and thus extract the hydraulic conductivity insidethe CNT. This hydraulic conductivity, corresponding to the permeability of the CNT withoutentrance/exit effects, is compared to the modified HP relationship in the study by Thomas andMcGaughey [29]. The enhancement factor is also calculated by dividing the hydraulic conductivityover a no-slip HP relationship in the absence of entrance/exit effects [29].

Entrance/exit effects will be negligible when frictional loss along the tube wall is dominant. In the caseof CNTs, the smooth surface and hydrophobicity provide low frictional loss, and these characteristics areknown to be the major factors responsible for the fast mass transport rate through CNTs [12]. Becausethe frictional loss inside CNTs is very low, the entrance/exit effects become dominant, especially for shortCNTs. Therefore, in understanding the relationship between CNT dimensions (radius and length) andwater flow, it is necessary to factor in the entrance/exit effects. Our study aims to comprehensivelyexplain the water permeability and flow enhancement of CNTs and the dependence of these character-istics on CNT radius and length, including the consideration of entrance/exit effects. Using MDsimulations, we obtained the flow rate and permeation coefficient of water flowing through CNTswith varying radii and lengths. The CNT radii used were approximately 0.813–2.046 nm, and the lengthswere about 5–60 nm. A water transport model including the entrance/exit effects was derived andcompared to the measured values. Furthermore, we recalculated the enhancement factor by factoring in

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the entrance/exit effects, thereby quantifying the efficiency of CNTs and establishing a relationshipbetween the changes in efficiency and the dimensions of CNTs, particularly the radius and length.

Methods

Pressure-Driven flow simulation

Figure 1 shows the simulation setup. CNTs with various radii were tested: (12,12), (16,16), (20,20),(24,24), (27,27), (30,30), and (34,34) armchair CNTs. The corresponding center-to-center radii were

0.813–2.046 nm, given by Rc�c ¼ffiffiffi3

pn

2π d, where Rc-c is the center-to-center CNT radius, n is the chiralindex, and d is the lattice constant of 0.246 nm. The CNT radius, R, used in this work was defined bysubtracting the width of depletion layer (~0.27 nm) near the CNT surface from the center-to-centerCNT radius [12, 32]. In this way, the effective radius accessible to water molecules was used, and thecalculations of flow rate and permeability were more accurate. The effective CNT radius, R, rangedfrom 0.55 to 2.05 nm. The CNT lengths (L) varied from 5 to 60 nm. The water bath sizes in the x andy dimensions (Lx; LyÞ ranged from 4 to 8 nm depending on the CNT radius in order to keep theporosity constant. The water bath size in the z dimension ðLzÞ was fixed at 6 nm.

Simulations were performed using the LAMMPS package [33]. A time step of 1 fs was used. A Nosé-Hoover thermostat [34] was used to maintain the temperature at 300 K with a time constant of 0.1 ps. Aperiodic boundary condition was applied in all three dimensions. A particle–particle–particle–meshmethod [35] was used for long-range electrostatic interactions. An SPC/E (extended simple point charge)water model [36] was used with a shake algorithm to constrain the angle and bond length. The SPC/Ewater model was used because various water properties, such as the diffusion coefficients [37], viscosities[38], and surface tension [39], were in good agreement with the experimental values. The carbon atomswere modeled with Lennard-Jones interactions with the parameters σC-C = 0.339 nm and εC-C = 0.0692kcal/mol [40]. The Lennard-Jones parameters between water and carbon atoms were obtained fromLorentz-Berthelot mixing rules: σC-O = 0.328 nm and εC-O = 0.104 kcal/mol.

Water permeability measurement and modeling

The water permeability of CNTs with varying lengths and radii was obtained by performing none-quilibrium molecular dynamics (NEMD) simulations. By applying a pressure drop, water flows throughthe CNTs were successfully simulated. The water flow rate was calculated by directly counting the netnumber of water molecules transported through the CNT during the simulation. The CNT permeabilitywas then obtained using pQ ¼ Q=ΔP, where pQ is the permeation coefficient, Q is the volumetric flowrate, and ΔP is the applied pressure drop. The pressure drops employed were 10–300 MPa. Afterconfirming the linear relationship by plotting the respective pressure drop and volumetric flow rate,the permeability was obtained from the linear slope. The applied pressure was very high compared to the

Figure 1. Simulation setup for water flow through a CNT. R and L are the CNT radius and length, respectively. Ly and Lz are thedimensions of the water bath in the y and z directions, respectively.

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experimental operating pressure (~10 MPa) [41]; this was to reduce statistical error caused by the highthermal motions of water molecules during the limited simulation time. The permeability obtained in thehigh pressure range was reported to be the same as that obtained in standard operating pressure range[4]. The linearity between the water flow rate and the pressure drop was shown in a wide pressure range,up to 500 MPa during the MD simulation [4, 42, 43].

The relational expression for modeling was based on hydrodynamic equations and circuitanalogy. When plotting the pressure along the length direction, as shown in Figure 2, the pressuregradient varied in the interior and at the entrance/exit points. By comparing the pressure to voltageand the flow rate to current, it was shown that the entrance/exit resistance and the interior resistancewere serially connected. Therefore,

Q ¼ 1ER þ IR

ΔP (1)

and

pQ ¼ 1ER þ IR

; (2)

where ER is the resistance in the entrance/exit region and IR is the resistance in the interior region.The interior resistance can be calculated using the Navier-Stokes equation. Instead of the no-slip

boundary condition, wherein the fluid velocity is assumed to be zero along the CNT wall, a slipboundary condition needs to be applied. Because of the weak water–carbon interaction (hydropho-bicity) and smooth surface, strong slip behavior at the CNT wall was observed [12]. The slipboundary condition is given by

u Rð Þ ¼ �δdudr

� �r¼R

; (3)

where u is the water velocity, R is the CNT radius, and δ is the slip length.After applying the slip boundary condition, the water velocity was calculated as shown below in

Eq. (4):

u rð Þ ¼ � 14μ

r2 þ R2

4μþ R2μ

δ

� �dPdz

: (4)

Figure 2. Pressure distribution along the CNT axial direction during pressure-driven flow simulations. Local pressure is calculatedusing the Irving-Kirkwood expression [44]. The applied pressure drops were 100, 200, and 300 MPa. The CNT membrane waspositioned between z = 6 and 21 nm. The pressure drop was large at the entrance and exit regions and small in the interiorregion.

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Here, r refers to the dimension of the CNT in the radial direction; at the center of the pore, r =0, and at the CNT wall, r = R. The variable z refers to the dimension in the length direction ofthe CNT, and μ refers to the dynamic viscosity of water. Upon integrating from r = 0 to r = R, a

volumetric flow rate similar to that of the HP equation was obtained from Q ¼ �0R u rð Þ2πrdr.This is shown as Eq. (5):

Q ¼ π R4 þ 4R3δð Þ8μL

ΔP: (5)

The interior resistance is denoted as in Eqs. (6) and (7):

IR ¼ iRL (6)

and

iR ¼ 8μπ R4 þ 4R3δð Þ ; (7)

where iR is the interior resistivity.The entrance resistance can be obtained using the flow resistance of an infinitely thin orifice. This

can be expressed by using the Sampson equation [45]:

ER ¼ 3μR3

: (8)

An infinite-series solution for a fluid moving though finite-length pores was obtained by Daganet al. [46]. They found that a simple approximate solution for combining Poiseuille flow at theinterior with Sampson flow at the entrance/exit was close to the exact solution within 1%error [46].

Calculation of viscosity and slip length

To obtain the entrance/exit resistance and interior resistance using Eqs. (6), (7), and (8), accuratevalues of the viscosity and slip length depending on the CNT radius are required. When watermolecules are confined in nanosized tubes, water properties such as diffusivity and viscosity deviatefrom their bulk values due to the change in water structure [47, 48]. The slip length is also influencedby the CNT radius [28, 49]. For an accurate calculation of the resistance that factors in the CNTradius, the values of the viscosity and slip length should be obtained in advance.

With the water bath removed, the periodic boundary condition was applied to calculate theviscosity and slip length inside the CNT, thus allowing the simulation of an infinitely long CNT. Thecalculations of viscosity and slip length were carried out using the Green-Kubo equations [32, 50] inan equilibrium MD simulation without any external forces. Some studies have calculated theviscosity and slip length indirectly by fitting the flow velocity profile obtained from NEMD to atheoretical model such as Eq. (4) [28, 51]. However, in this study, this method was not adopted forseveral reasons. First, the velocity profile is flat because of the relatively large slip length compared tothe CNT radius. In this case, a huge error may occur in the obtained viscosity and slip length [52].Second, when determining the validity of our model by comparing the NEMD result with themodeling result, the judgment would have been compromised by using results obtained from anidentical simulation. An independent equilibrium MD simulation was conducted in order to avoidrelating the modeling result to the data points based on NEMD.

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Results and analysis

Viscosity and slip length variation with CNT radius

The viscosity and slip length were calculated for each CNT radius, and they are shown in Figure 3.As shown in Figure 3a, the viscosity of the water inside the CNT was lower than the bulk value.The viscosity also decreased with decreasing CNT radius. At the hydrophobic surface of the CNT,the number of hydrogen bonds among the water molecules was reduced, and “dangling” OHbonds were observed [12]. The reduced amount of hydrogen bonding of the water moleculesresulted in an increase in the degree of freedom of each water molecule, which led to an increase inthe diffusion coefficient and a decrease in the viscosity. A similar mechanism of enhanceddiffusion was observed at the liquid–vapor interface [53]. At the interfacial region, a 58% greaterdiffusion coefficient than that in the bulk water was observed due to the breaking up of the bulkliquid hydrogen bonding [53].

The changes in viscosity caused by changes in the CNT radius can be expressed as the fractionalaverage of the water viscosity at the surface region and the bulk water viscosity of the correspondingarea, as shown in Eq. (9) [28]. Considering the surface viscosity to be 0.32 mPa∙s and the surfacelayer thickness to be 0.38 nm, Eq. (9) well describes the changes to viscosity, which are shown inFigure 3a as a solid line.

μ Rð Þ ¼ μsAs Rð ÞAT Rð Þ þ μbulk 1� As Rð Þ

AT Rð Þ� �

: (9)

Figure 3. Change in (a) viscosity and (b) slip length against CNT radius. As the radius increases, the viscosity approximates the bulkvalue (0.855 mPa∙s), and the slip length approximates the case of a flat graphene surface (66.1 nm). These limit values are depictedusing a blue dashed line. The solid red lines in (a) and (b) depict Eqs. (10) and (11), respectively.

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Here, AT and As refer to the total cross-sectional area and the cross-sectional area at the surfaceregion inside the CNT, respectively. In other words, Ap ¼ πR2 and As ¼ πR2 � π R� 0:38nmð Þ2.The variable μbulk is the viscosity of the bulk water (0.855 mPa∙s at 300 K [54]), and μs is the viscosityat the surface region (0.32 mPa∙s). Rewriting Eq. (9) using the relational expression for the CNTradius,

μ Rð Þ ¼ C1

R2þ C2

Rþ μBulk: (10)

Here, C1 = 0.075 mPa∙s∙nm2 and C2 = −0.4 mPa∙s∙nm.Figure 3b plots the slip length against the CNT radius. The power law was applied to the variation

in slip length with CNT radius as follows:

δ Rð Þ ¼ ARB þ δR!1; (11)

where δR!1 is the slip length for an infinite pore radius, R is the CNT radius in nanometers, and Aand B are power law coefficients, which were determined using the least squares method. Theirvalues were 21.06 nm2.36 and −1.36 (dimensionless), respectively. The value of δR!1 is obtainedfrom the slip length of the flat graphene surface to capture infinitely small curvature as R ! 1. Thecomputed slip length of the graphene surface (δR!1) was 66.1 nm, which falls between the reportedvalues of 30 nm computed by Thomas and McGaughey [28] and 80 nm computed by Falk et al. [49].The obtained slip length of 66.1 nm was also close to the value of 60 nm reported by Kannam et al.[52] and 64 nm by Koumoutsakos et al. [55]. A summary of the slip lengths for CNTs with variousradii and a planar graphene surface can be found in Kannam et al. [56].

Equation (11) is plotted in Figure 3b as a solid line. The increasing pore curvature has the effect ofsmoothing the energy landscape, which can reduce the surface friction coefficient (λ ¼ μ=δÞ [49].The variation in the friction coefficient with the pore radius is much larger than the variation in theviscosity with the pore radius. Thus, the slip length increases with a decreasing pore radius, followingthe trend of the friction coefficient. Note that the weak interaction energy between water and carbon(hydrophobicity) and a smooth pore surface contribute to a large slip length by forming a depletionlayer [12]. The zero-curvature case of the flat graphene surface already shows a large slip length(~66 nm).

To determine the accuracy of the calculated slip length and viscosity, the velocity profile wasobtained by substituting these factors into Eq. (4). The velocity profiles obtained from each pressuregradient cases are shown in different colors in Figure 4; the solid line indicates Eq. (4), and the datapoints indicate the water velocities measured directly from the NEMD simulation. As shown inFigure 4, the velocity profiles are in close agreement with each other, indicating that the slip lengthand viscosity obtained using the Green-Kubo equation accurately represent the water dynamics insidea CNT. The flat water velocity profile indicates that a large slip length occurs at the CNT wall [12].

Change in water permeability with CNT radius and length

The permeability of a CNT depends on its radius and length, so to investigate this phenomenon, wevaried the radius and length. First, the CNT length was fixed at 10 nm, and the CNT radius was adjustedfrom 0.5 to 2 nm. The measured permeability from NEMD simulations is represented by the points inFigure 5. As mentioned previously, the permeation coefficient is obtained from the slope of the linear fitbetween the water flow rate and the applied pressure. The error bars in Figure 5 indicate the standarderrors of the linear fit coefficients. The modeling results of Eqs. (2) and (6) through (8) are representedby solid lines. The viscosity and slip-length models (Eqs. (10) and (11)) obtained from the previoussection are also applied to take into account their variation with CNT radius.

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When applying the bulk viscosity value without considering the changes to the structure of the watermolecules and their properties caused by nanoscale confinement inside a CNT [57, 58], the resistance wasoverestimated compared to the case that applied the actual viscosity values inside the CNT. The over-estimate can result in a large difference between the predicted value using the flow model and the actualmeasured value for a small CNT radius. As shown in Figure 5, the water flowmodel represents the variationin permeability with CNT radius reasonably well when properly assessing the entrance resistance andinterior resistance by including the viscosity and slip length variations that depend on the radius.

Next, using a fixed CNT radius, the permeability changes due to changes in CNT length wereobserved in (20,20) and (30,30) CNTs. With an increasing length of the CNT up to 60 nm, thepermeability measured through NEMD simulation is represented by the round points in Figure 6.The modeling results of Eqs. (2) and (6) through (8) are represented by solid lines. Approximately,

Figure 4. Water velocity profile in the radial direction of (20,20) CNT. Due to the influence of large slip length at the surface andthe nanoscale radius, the velocity profile resembles a plug-like shape rather than a parabolic shape.

Figure 5. Water permeability plotted against CNT radius. Data points were obtained from NEMD, and the red solid line shows themodeling results using Eq. (2).

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the changes in permeability over a length of 5 nm were well predicted by our proposed model. Belowa length of 5 nm, the prediction deviated due to the coupling between the entrance and exit regions.When the entrance and exit regions are overlapped, water molecules form a layered structure in theaxial direction, and the physical properties of the water change according to its structure [32].

Note that the permeability changes with CNT length are relatively small compared to thestatistical error of the MD simulations. The water flow rate or permeability variations with lengthmay not be properly captured if narrow length ranges are tested [26, 27]. Simulations of CNTs withmicrometer-scale lengths are computationally expensive, especially for CNTs with a large radius,because a large number of water molecules are needed to fill the CNT at the density of liquid water.Despite the high computational cost, a simulation of a CNT up to 2 μm in length was attempted byWalther et al. with a (15,15) CNT [58]. In our study, it was shown that the water flow model predictsthe water flow rate and the permeability of a CNT reasonably well; the analysis for CNTs withmicrometer-scale lengths can be performed using the water flow model without a high computa-tional expense.

Entrance and exit effects of CNTs

We examined the entrance/exit effects and surface effects of CNTs by modulating the slip length inthe water flow model. The changes in water permeability when there are only entrance/exit effects(no surface effects; δ ¼ 1) are illustrated in Figure 7. The solid red line shows the perfect slip case,which has only entrance/exit effects. In the perfect slip case, the interior resistance (Eq. (7)) is zerowhen the infinite slip length is applied. In this case, the permeation coefficient is reduced to 1/ER,and there are no changes in permeability with length.

The water permeability changes under no-slip surface conditions (maximum surface effect;δ ¼ 0Þ are also illustrated in Figure 7. The solid purple line shows the case when no-slip surfaceconditions occur, plotted by applying the zero-slip length in Eq. (7). The solid yellow line shows thepartial-slip case, which refers to the hydrophobicity of CNTs, plotted by applying a slip length of

Figure 6. Variation in water permeability with CNT length. Data points were obtained through NEMD simulation, and the solid redlines show the modeling results of Eq. (2). As the CNT length increases, the permeability gradually decreases. This is due to theincrease in interior resistance with the increase in CNT length.

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84.93 nm corresponding to the (20,20) CNT in Eq. (7). NEMD data points for a (20,20) CNT areshown for reference in Figure 7.

In partial-slip or no-slip cases, the interior resistance increases proportionally to the tube length.Therefore, as the length increases, the permeability decreases. In the no-slip case, the rate of decreaseis very high, whereas the rate of decrease is moderate in partial-slip case such as CNTs (only an~13% reduction for a 50-nm-long CNT). Therefore, in the case of a membrane with a nanoscale

Figure 7. Entrance/exit effects and surface effects on water permeability variation with CNT length. Permeability is largely reducedin the no-slip surface conditions corresponding to a hydrophilic surface. In case of CNTs, the reduction is small due to the slipsurface.

Figure 8. Fraction of entrance resistance [ER/(ER + IR)]. The percentage of entrance resistance decreases with CNT length startingfrom 100%. (Inset) The critical length is obtained from the length when the percentage of entrance resistance reaches 5% of thetotal resistance. Above this critical length, the entrance/exit effects can be neglected.

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thickness, the entrance/exit effects rather than the surface effects should be considered. In addition,it shows that the slip surface condition of CNTs is a major factor in the high permeability of CNTs.

In the analysis of water flow through CNT membranes, the entrance/exit resistances have seldombeen considered [2, 12, 28, 29]. In the case of a hydrophilic surface where no surface slip occurs, theentrance/exit resistance is negligible because the interior resistance is relatively much higher than theentrance/exit resistances. However, when the interior resistance is low, as for a CNT membrane, theentrance/exit resistances are important. Therefore, the length criteria, which can ignore the entrance/exit resistances, were estimated by extrapolating the modeling results and observing the ratiobetween the entrance/exit resistances and the total resistance up to a microscale length (see Figure8). The length criteria are determined when the entrance/exit resistances are lower than 5% of theentire resistance—in our case, mostly determined to be between 5 and 20 µm at a pore radius of lessthan 2 nm.

Recalculation of CNT enhancement factor

Finally, the enhancement factor, which is a figure that quantitatively represents the efficiency of aCNT membrane, was recalculated by considering the entrance/exit effects. The enhancement factoris defined as [2]

E ¼ QCNT=QHP: (12)

Here, E is the enhancement factor, QCNT is the measured water flow rate through the CNT, and QHP

is the water flow rate according to the Hagen-Poiseuille equation. Without considering the entrance/exit resistances, QCNT can be expressed by Eq. (5). Therefore, the enhancement factor withoutentrance/exit effects is as follows [28]:

E ¼ 1þ 4δR

� �μBulkμCNT

: (13)

Equation (13) is independent of the CNT length by excluding the entrance/exit effects.Considering the entrance/exit resistances, the HP flow model should also be corrected. The flow

rate relational equations for each case are arranged in Table 1, and the corrected enhancement factorcan be finally expressed as follows:

Ecorr ¼ QCNT;corr

QHP;corr¼ 1þ 32Lδ

3πR2 þ 8LRþ 12πδR

� �μBulkμCNT

� �: (14)

In addition to the water flow rate Q, as shown in Eq. (14), the enhancement factor is also afunction of both the length and radius of the CNT. Moreover, the dependence of the slip length andviscosity on the CNT radius indirectly affects the variation of enhancement with CNT radius.

The corrected enhancement factor (Eq. (14)) obtained from the suggested model, the valueobtained from the NEMD of the present study, and the published values obtained from either

Table 1. Water flow model for each case and enhancement factor.

Without entrance/exit effects With entrance/exit effects

No-slip and bulk viscosity QHP ¼ πR48μBulkL

ΔP QHP;corr ¼ 8μBulkLπR4 þ 3μBulk

R3

� ��1ΔP

Slip and reduced viscosity QCNT ¼ πðR4þ4R3δÞ8μCNTL

ΔP QCNT;corr ¼ 8μCNTLπ R4þ4R3δð Þ þ 3μCNT

R3

� ��1ΔP

Enhancement factor E ¼ 1þ 4δR

μBulkμCNT Ecorr ¼ 1þ 32Lδ

3πR2þ8LRþ12πδR

� �μBulkμCNT

� �

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simulations or experiments were compared. Many CNT membranes are being experimentallymanufactured and tested, and some of the results include permeation not only into the CNT butalso into the polymer matrix [14]. Such cases were excluded in our comparison, because this studyconsiders only the permeation through the inside of the CNT. The CNT–wall membranes fabricatedrecently were also excluded in the comparison, because the water flows through both the inside andthe outside of the CNT. A theoretical study of water flow through in between CNTs can be found inTajiri et al. [11].

First, the enhancement factor changes due to the length change were examined. The experiment/simulation results with radii of ~0.8 and ~1.7 nm and the theoretical estimation using Eq. (14) areshown in Figures 9a and 9b. When the CNT length is short, the entrance/exit effects are relativelystrong. In this case, the enhancement factor is not very large. As the CNT length increases, so doesthe total area of the hydrophobic surface within the CNT. Compared to the HP flow of the same

Figure 9. Enhancement factor variations with CNT length. (a) Comparisons of published data and theoretical models for CNT radiiranging from 0.6 to 1.0 nm. The red diamonds represent simulation results obtained from the current work using the LAMMPSprogram [33], using a CNT radius of 1.09 nm. The open circles show simulation results obtained from Walther et al.’s work [58]with a CNT radius of 0.75 nm. The blue and red circles are the results from the NAMD program with pressure drops of 20 and 2MPa, respectively. The green circles show the results from the FASTTUBE program. The brown squares are experimental resultsobtained from Holt et al.’s work [2] with an average CNT radius estimated as ~0.8 nm. (b) Comparisons of published data and atheoretical model for a CNT with a radius of ~1.7 nm. The red diamonds show simulation results obtained from current work witha CNT radius of 1.77 nm. The brown squares show experimental results obtained from Bui et al.’s work [7], with an average CNTradius estimated as 1.65 nm. The blue circles show experimental results obtained from Kim et al.’s work [13] with an average CNTradius estimated as 1.65 nm.

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length, the enhancement factor also increases consequentially as the water flux increases. When L →

∞, Eq. (14) approximates Eq. (13). In other words, this case does not need to consider the entrance/exit effects, and the enhancement factor asymptotes to the single value, which lies between 100 and10,000 for a radius < 2.5 nm.

The published values usually show a large uncertainty, possibly caused by the experimentalchallenges associated with accurately calculating the porosity based on open CNTs, measuringthe CNT radius, and measuring the distribution of the CNT radii [25]. Despite large uncertain-ties among different sets of experimental data, Eq. (14) represents the trend of variation in theenhancement factor reasonably well. Moreover, Eq. (14) shows the best match to the results ofBui et al. [7], which were recently presented (Figure 9b, brown square).

The changes in the enhancement factor with the radius are shown in Figure 10 for the casesL = 10 nm, 100 nm, 1 μm, and approaching infinity as the most representative. Similarly, thevalues obtained from an MD simulation of the present study and the values obtained fromexperiments were compared. The case of L → ∞ corresponds to the upper limit of theenhancement factor. The experimental values presented by Holt et al. [2] show a widerange, and the value of the lower bound is within the theoretical range. The experimentalvalues presented by Qin et al. [59] are usually lower than the theoretical values whenconsidering a micrometer-scale length. Qin et al. [59] showed the possibility of uncertaintywhen estimating the driving force [25]. Holt et al. [2] showed the possibility of uncertaintywhen estimating the open CNT number [25]. Although published results show large devia-tions, the trend in variation of the enhancement factor is described by our proposed model(Eq. (14)) reasonably well.

Figure 10. Variations in enhancement factor with CNT radius. The red diamonds show simulation results obtained from the currentwork with a CNT length of 0.01 μm. The brown squares show experimental results obtained from Qin et al.’s work with a 280-μmCNT length [59]. The green circles show experimental results obtained from Bui et al.’s work with a 23-μm CNT length [7]. Thepurple triangles show experimental results obtained from Kim et al.’s work with a CNT length of 20–50 μm [13]. The blue invertedtriangles show experimental results obtained from Holt et al.’s work with a CNT length of 2–3 μm [2].

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Conclusions

In this study, the relationship between the water flow rate through CNT membranes and theCNT dimensions (radius and length) was examined with consideration of the entrance/exiteffects. Molecular dynamics simulations were performed to measure the water flow rate throughCNTs with radii of approximately 0.813–2.046 nm and lengths of 5–60 nm. The obtained waterflow rate was compared to a flow model based on continuum hydrodynamics and a circuitanalogy. The entrance/exit resistance and the interior resistance were obtained from a modifiedHagen-Poiseuille’s equation with a slip boundary condition and the Sampson equation, respec-tively. In the application of the flow model, the variations in water viscosity and slip length withthe CNT radii were also calculated and applied. The derived water flow model was in goodagreement with the measured flow rate using molecular dynamics simulations. Based on thewater flow model, the critical tube length for the consideration of entrance/exit effects wasestimated to be about 5–20 µm for a pore radius of about 0.5–2 nm. The enhancement factorwas also recalculated considering the entrance/exit effects, and it showed a decrease due to therelative increase in the entrance/exit effects as the CNT length decreased. Through comparisonwith the simulation/experimental values reported from previous studies, it was revealed that thesuggested model could be useful for predicting and analyzing the water flow inside the CNTmembrane.

Funding

The improvement of this research was supported by a grant from the R&D program of the Korea Railroad ResearchInstitute.

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